Compound Measures

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수업 노트

Compound Measures Overview

A compound measure uses more than one measurement (e.g., speed = distance ÷ time).
Compound measures often describe rates – how one quantity changes per unit of another.
Examples: speed, density, pressure, flow rate, population density, fuel consumption.
Units are derived from the formula, e.g., km/h means distance (km) per time (h).
Use formula triangles to rearrange: cover the variable you need, then multiply or divide the other two.

Speed, Distance & Time

Speed = distance ÷ time (average speed over a journey).
Common units: m/s or km/h.
Distance = speed × time; Time = distance ÷ speed.
Check units: if distance is in km and time in minutes, speed is km/min.
For mixed units (e.g., minutes and hours), convert to consistent units before calculating.
In multi‑part journeys, treat each part separately or combine total distance and total time.

Speed, Distance, Time Formula Triangle

Density, Mass & Volume

Density = mass ÷ volume.
Common units: g/cm³ or kg/m³.
Mass = density × volume; Volume = mass ÷ density.
An object floats if its density is less than the density of water (1 g/cm³).
Ensure consistent units: convert kg to g or m to cm as needed.

Density, Mass, Volume Formula Triangle

Pressure, Force & Area

Pressure = force ÷ area.
Common units: N/m² (also called Pascals, Pa).
Force = pressure × area; Area = force ÷ pressure.
Remember: weight is a force (measured in newtons), not mass.
Area units must match: if force is in N and area in m², pressure is N/m².

Pressure, Force, Area Formula Triangle

Unit Conversions

Always check if the answer requires different units (e.g., m/s to km/h).
To convert km/h to m/s: multiply by 1000/3600 (or divide by 3.6).
To convert m/s to km/h: multiply by 3600/1000 (or multiply by 3.6).
For time: 1 hour = 60 minutes = 3600 seconds.
For mass: 1 kg = 1000 g; for length: 1 km = 1000 m; for volume: 1 m³ = 1 000 000 cm³.

Worked Examples – Speed

Example: George drives 315 km in 3 h 30 min. Average speed = 315 ÷ 3.5 = 90 km/h.
Example: A speedboat travels 84 km/h. Convert to m/min: 84 × 1000 ÷ 60 = 1400 m/min.
Example: Ivan walks 1.5 km in 20 min. Speed = 1.5 ÷ (20/60) = 4.5 km/h.
Example: A car travels at 20 m/s for 10 km. Time = 10 000 ÷ 20 = 500 s = 8 min 20 s.

Worked Examples – Density & Pressure

Density example: Wood block 8 × 20 × 10 cm, mass 1360 g. Volume = 1600 cm³, density = 1360/1600 = 0.85 g/cm³ (floats).
Pressure example: Force 300 N on area 5 m². Pressure = 300/5 = 60 N/m².
Pressure example: Force 300 N, pressure 0.6 N/cm², width 20 cm. Area = 300/0.6 = 500 cm², length = 500/20 = 25 cm.
Always state units in final answer.

Exam Tips

Memorise the three formulas: speed, density, pressure.
Watch for mixed units (e.g., minutes and hours) – convert before calculating.
Use formula triangles to avoid rearranging errors.
Show all working clearly, especially when converting units.
For floating questions, compare density to water (1 g/cm³).

슬라이드

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연습 문제

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1.A car travels 180 km in 3 hours. What is its average speed in km/h?
Easy
- A60 km/h
- B540 km/h
- C30 km/h
- D90 km/h
2.What is the formula for pressure?
Easy
- AP = F / A
- BP = F × A
- CP = A / F
- DP = F + A
3.A substance has a mass of 50 g and a volume of 25 cm³. What is its density?
Easy
- A2 g/cm³
- B0.5 g/cm³
- C1250 g/cm³
- D75 g/cm³
4.A force of 200 N is applied over an area of 4 m². What is the pressure?
Easy
- A50 N/m²
- B800 N/m²
- C0.02 N/m²
- D200 N/m²
5.George drives 315 km in 3 hours 30 minutes. What is his average speed in km/h?
Medium
- A90 km/h
- B105 km/h
- C80 km/h
- D110.25 km/h
6.A speedboat travels at 84 km/h. Convert this speed into metres per minute.
Medium
- A1400 m/min
- B5040 m/min
- C1.4 m/min
- D84000 m/min
7.Ivan walks 1.5 km in 20 minutes. What is his average speed in km/h?
Medium
- A4.5 km/h
- B3 km/h
- C0.075 km/h
- D30 km/h
8.A cuboid block of wood measures 8 cm × 20 cm × 10 cm and has mass 1360 g. The density of water is 1 g/cm³. Will the block float?
Hard
- AYes, because its density is 0.85 g/cm³
- BNo, because its density is 1.18 g/cm³
- CYes, because its density is 1 g/cm³
- DNo, because its density is 0.85 g/cm³

더 많은 문제

9.A rectangle is 0.8 m wide and 1.2 m long. A force F produces a pressure of 200 N/m². Find F.
Medium
- A192 N
- B0.192 N
- C200 N
- D1920 N
10.A force of 300 N is exerted on a rectangle of width 20 cm and length x cm. The pressure is 0.6 N/cm². Find x.
Hard
- A25 cm
- B9 cm
- C0.04 cm
- D250 cm
11.A cuboid block of wood measures 0.2 m × 0.2 m × 0.1 m and has mass 3 kg. Density of water is 1 g/cm³. Will it float?
Hard
- AYes, because its density is 0.75 g/cm³
- BNo, because its density is 1.33 g/cm³
- CYes, because its density is 0.075 g/cm³
- DNo, because its density is 0.75 g/cm³
12.What are the units of density if mass is in kilograms and volume is in cubic metres?
Easy
- Akg/m³
- Bm³/kg
- Ckg m³
- Dkg per m
13.The distance between Madrid and Buenos Aires is 10050 km. Diego's flight takes 12 hours 30 minutes. Calculate the average speed in km/h.
Medium
- A804 km/h
- B837.5 km/h
- C800 km/h
- D670 km/h
14.Kiran leaves home at 9:45 am, drives 135 km, and arrives at 11:15 am. What is her average speed in km/h?
Medium
- A90 km/h
- B67.5 km/h
- C135 km/h
- D45 km/h
15.A plane flies from Melbourne to Tokyo at 783 km/h. Distance is 8352 km. It leaves Melbourne at 09:52 local time. Tokyo time is 2 hours behind Melbourne. Find the local time in Tokyo when it arrives.
Challenge
- A18:32
- B20:32
- C16:32
- D22:32
16.A car travels 150 km in 2.5 hours. What is its average speed?
Easy
- A60 km/h
- B50 km/h
- C75 km/h
- D45 km/h
17.A substance has a mass of 80 g and a volume of 20 cm³. What is its density?
Easy
- A4 g/cm³
- B0.25 g/cm³
- C1600 g/cm³
- D100 g/cm³
18.A force of 500 N is applied over an area of 2 m². What is the pressure?
Easy
- A250 N/m²
- B1000 N/m²
- C0.004 N/m²
- D502 N/m²
19.A runner covers 400 m in 50 seconds. What is her average speed in m/s?
Easy
- A8 m/s
- B0.125 m/s
- C20 m/s
- D350 m/s
20.A cyclist travels 18 km in 45 minutes. What is her average speed in km/h?
Medium
- A24 km/h
- B13.5 km/h
- C0.4 km/h
- D36 km/h
21.A rectangular metal plate has area 0.5 m². A force of 150 N produces a pressure of 300 N/m². What is the force?
Medium
- A150 N
- B600 N
- C75 N
- D300 N
22.A block of wood has mass 600 g and volume 750 cm³. Will it float in water? (Density of water = 1 g/cm³)
Medium
- AYes, because its density is 0.8 g/cm³
- BNo, because its density is 1.25 g/cm³
- CYes, because its density is 1.25 g/cm³
- DNo, because its density is 0.8 g/cm³
23.A car travels at 30 m/s. How long does it take to travel 9 km? Give your answer in minutes.
Medium
- A5 minutes
- B300 minutes
- C0.3 minutes
- D50 minutes
24.A force of 200 N is applied to a rectangle of width 0.5 m and length 0.8 m. What is the pressure in N/m²?
Medium
- A500 N/m²
- B80 N/m²
- C0.002 N/m²
- D200.5 N/m²
25.A liquid has density 0.8 g/cm³. What is the mass of 250 cm³ of this liquid?
Medium
- A200 g
- B312.5 g
- C0.0032 g
- D250.8 g
26.A plane flies 2400 km in 3 hours 20 minutes. What is its average speed in km/h?
Easy
- A720 km/h
- B800 km/h
- C600 km/h
- D750 km/h
27.A cube of side 4 cm has mass 256 g. What is its density in g/cm³?
Easy
- A4 g/cm³
- B16 g/cm³
- C0.25 g/cm³
- D64 g/cm³
28.A pressure of 50 N/m² is exerted by a force of 20 N. What is the area in m²?
Easy
- A0.4 m²
- B2.5 m²
- C1000 m²
- D70 m²
29.A car travels at 25 m/s for 2 minutes. How far does it travel in km?
Medium
- A3 km
- B0.83 km
- C50 km
- D0.05 km
30.A cuboid has dimensions 5 cm × 4 cm × 3 cm and mass 180 g. What is its density?
Medium
- A3 g/cm³
- B0.333 g/cm³
- C12 g/cm³
- D1.5 g/cm³
31.A rectangular tank measures 1.2 m by 0.8 m by 0.5 m. It is filled with water. The density of water is 1000 kg/m³. What is the mass of water in the tank?
Medium
- A480 kg
- B4800 kg
- C48 kg
- D4.8 kg
32.A car travels at a constant speed of 25 m/s. How many kilometres does it travel in 1 hour 20 minutes?
Medium
- A120 km
- B90 km
- C100 km
- D108 km
33.A force of 240 N is applied to a rectangular surface of width 0.6 m and length 0.4 m. What is the pressure in N/m²?
Medium
- A1000 N/m²
- B100 N/m²
- C10000 N/m²
- D10 N/m²
34.A liquid has density 1.2 g/cm³. What is the mass of 50 cm³ of this liquid?
Easy
- A60 g
- B41.7 g
- C600 g
- D0.024 g
35.A cylinder has radius 7 cm and height 10 cm. It is made of metal with density 8 g/cm³. What is the mass of the cylinder? (Take π = 22/7)
Medium
- A12320 g
- B1760 g
- C440 g
- D1232 g
36.A pressure of 500 N/m² is exerted by a force of 200 N. What is the area in m²?
Easy
- A0.4 m²
- B2.5 m²
- C0.04 m²
- D4 m²
37.A cyclist travels 15 km at 20 km/h, then another 10 km at 30 km/h. What is her average speed for the whole journey?
Hard
- A23.1 km/h
- B25 km/h
- C22.5 km/h
- D24 km/h
38.What is the formula for density?
Easy
- ADensity = Mass / Volume
- BDensity = Volume / Mass
- CDensity = Mass × Volume
- DDensity = Mass + Volume
39.A block of wood has mass 800 g and volume 1000 cm³. Will it float in water? (Density of water = 1 g/cm³)
Medium
- AYes, because its density is 0.8 g/cm³
- BNo, because its density is 1.2 g/cm³
- CYes, because its density is 1 g/cm³
- DNo, because its density is 0.8 g/cm³
40.The distance between two cities is 450 km. A train leaves at 08:30 and arrives at 12:00. What is its average speed in km/h?
Medium
- A128.6 km/h
- B150 km/h
- C112.5 km/h
- D90 km/h
41.Which of the following is a compound measure?
Easy
- ASpeed
- BLength
- CMass
- DTime
42.What are the units of density when mass is in kilograms and volume is in cubic metres?
Easy
- Akg/m³
- Bkg/m²
- Cg/cm³
- DN/m²
43.Pressure is defined as force per unit area. Which formula is correct?
Easy
- AP = F × A
- BP = F / A
- CP = A / F
- DP = F + A
44.Speed is a compound measure because it involves both distance and time.
Easy
True or false?
45.A block of wood has mass 450 g and volume 600 cm³. The density of water is 1 g/cm³. Will the block float?
Medium
- AYes, because its density is less than water
- BNo, because its density is greater than water
- CYes, because its mass is less than water
- DNo, because its volume is less than water
46.Match each compound measure to its formula.
Medium
- Speed
- Density
- Pressure
- Mass / Volume
- Force / Area
- Distance / Time
47.Arrange the following steps to calculate the average speed for a journey of 120 km completed in 1 hour 30 minutes. (Order the steps correctly.)
Medium
- Convert 1 hour 30 minutes to 1.5 hours
- Write formula: speed = distance / time
- Substitute: speed = 120 / 1.5
- Calculate: speed = 80 km/h
48.Match each compound measure to its correct formula.
Medium
- Speed
- Density
- Pressure
- Mass / Volume
- Distance / Time
- Force / Area
49.Arrange the following steps in the correct order to calculate the average speed of a car that travels 150 km in 2 hours 30 minutes.
Medium
- Write the formula: speed = distance / time
- Convert 2 hours 30 minutes to 2.5 hours
- Substitute: speed = 150 / 2.5
- Calculate: speed = 60 km/h
50.Which of the following are compound measures? Select all that apply.
Easy
- ASpeed
- BMass
- CDensity
- DTime
- EPressure
51.Complete the sentence.
Easy
The formula for density is density = ____ / ____.
52.A force of 300 N is applied to an area of 2.5 m². Calculate the pressure in N/m².
Easy
53.A liquid has a density of 0.9 g/cm³. What is the mass of 200 cm³ of this liquid? Give your answer in grams.
Easy
54.A car travels at an average speed of 60 km/h for 2 hours 30 minutes. How many kilometres does it travel?
Medium
55.Write down the formula triangle for speed, distance and time. (Write the three variables in the correct positions.)
Easy
56.A force of 250 N is applied to an area of 0.5 m². Calculate the pressure in N/m².
Easy
57.Write down the formula for average speed in terms of distance and time.
Easy
58.A force of 300 N is applied to an area of 2.5 m². Calculate the pressure in N/m².
Medium
- A144 N/m²
- B120 N/m²
- C108 N/m²
- D132 N/m²
59.A liquid has a density of 0.9 g/cm³. What is the mass of 200 cm³ of this liquid? Give your answer in grams.
Medium
- A180 g
- B162 g
- C216 g
- D198 g
60.A car travels at an average speed of 60 km/h for 2 hours 30 minutes. How many kilometres does it travel?
Medium
- A135 km
- B180 km
- C150 km
- D165 km
61.A force of 250 N is applied to an area of 0.5 m². Calculate the pressure in N/m².
Medium
- A450 N/m²
- B600 N/m²
- C500 N/m²
- D550 N/m²

복습 노트

## Compound Measures - **Compound measures** use two or more measurements to calculate a rate (how one quantity changes per unit of another). - Common examples: speed, density, pressure, flow rate, population density, fuel consumption. - Units are derived from the formula: e.g., speed = distance/time → units: km/h or m/s. ### Speed, Distance & Time - **Formula:** \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \) (average speed). - Use formula triangle: \( D = S \times T \), \( S = D \div T \), \( T = D \div S \). - Watch for mixed units (e.g., minutes to hours, metres to km). ### Density, Mass & Volume - **Formula:** \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). - Units: g/cm³ or kg/m³. - An object floats if its density is less than water (1 g/cm³). ### Pressure, Force & Area - **Formula:** \( \text{Pressure} = \frac{\text{Force}}{\text{Area}} \). - Units: N/m² (Pascals). - Force (weight) is different from mass. ### Key Tips - Always check unit consistency before calculation. - Convert time: 1 hour = 60 minutes, 1 minute = 60 seconds. - Convert length: 1 km = 1000 m, 1 m = 100 cm. - For multi‑step problems, break into parts and apply the correct formula. ### Example (from past paper) - George drives 315 km in 3h 30min. Average speed = \( \frac{315}{3.5} = 90 \) km/h. - Pressure = Force/Area: 300 N on 5 m² → pressure = 60 N/m².

Compound measure: A measure calculated using more than one measurement, often a rate (e.g., speed, density, pressure).
Speed: Distance travelled per unit time; formula: Speed = Distance / Time.
Density: Mass per unit volume; formula: Density = Mass / Volume.
Pressure: Force per unit area; formula: Pressure = Force / Area.
Average speed: Total distance divided by total time taken.
Flow rate: Volume of fluid passing a point per unit time; formula: Flow rate = Volume / Time.
Population density: Number of people per unit area; formula: Population density = Number of people / Area.
Fuel consumption: Distance travelled per unit volume of fuel; formula: Fuel consumption = Distance / Volume.
Formula triangle: A diagram showing the relationship between three quantities in a compound measure; used to rearrange formulas.

플래시카드

What is a compound measure?

A compound measure is calculated using more than one measurement, e.g., speed, density, pressure.

Write the formula for speed.

Speed = Distance / Time

Write the formula for density.

Density = Mass / Volume

Write the formula for pressure.

Pressure = Force / Area

A car travels 315 km in 3 hours 30 minutes. Calculate its average speed.

Average speed = 315 km / 3.5 h = 90 km/h

Convert 84 km/h into metres per minute.

84 km/h = 84000 m / 60 min = 1400 m/min

Ivan walks 1.5 km in 20 minutes. Find his average speed in km/h.

20 min = 1/3 h, so speed = 1.5 km / (1/3 h) = 4.5 km/h

An object exerts 300 N on an area of 5 m². Find the pressure.

Pressure = 300 N / 5 m² = 60 N/m² (or 60 Pa)

Ramond walks 2460 m in 33 minutes. Find average speed in km/h.

2460 m = 2.46 km, 33 min = 33/60 h = 0.55 h, speed = 2.46 / 0.55 = 4.4727... ≈ 4.47 km/h

Kiran leaves at 9:45 am, drives 135 km, arrives at 11:15 am. Find average speed.

Time = 1.5 h, speed = 135 km / 1.5 h = 90 km/h

A car travels at 20 m/s. How long to travel 10 km? Give answer in minutes and seconds.

10 km = 10000 m, time = 10000 / 20 = 500 s = 8 minutes 20 seconds

A rectangle 0.8 m by 1.2 m has pressure 200 N/m². Find the force F.

Area = 0.8 × 1.2 = 0.96 m², F = Pressure × Area = 200 × 0.96 = 192 N

A wood block 8 cm × 20 cm × 10 cm has mass 1360 g. Density of water is 1 g/cm³. Will it float?

Volume = 8×20×10 = 1600 cm³, density = 1360/1600 = 0.85 g/cm³ < 1, so it floats.

A plane flies 8352 km at 783 km/h. Leaves Melbourne 09:52 local. Tokyo is 2 hours behind. Find arrival time in Tokyo.

Time = 8352/783 = 10.666... h = 10 h 40 min. Depart 09:52 Melbourne = 07:52 Tokyo. Arrival = 07:52 + 10:40 = 18:32 Tokyo time.

A car travels at 45 km/h for 5 minutes. Wheel radius 25 cm. Find number of complete revolutions.

Distance = 45 km/h × (5/60) h = 3.75 km = 375000 cm. Circumference = 2π×25 ≈ 157.08 cm. Revolutions = 375000 / 157.08 ≈ 2387.

A plane flies 375 km at 450 km/h. Leaves at 14:45. Find arrival time.

Time = 375/450 = 0.8333... h = 50 min. Arrival = 14:45 + 50 min = 15:35.

A force of 300 N on a rectangle of width 20 cm and length x cm gives pressure 0.6 N/cm². Find x.

Area = Force/Pressure = 300/0.6 = 500 cm². Area = 20 × x, so x = 500/20 = 25 cm.

A wood block 0.2 m × 0.2 m × 0.1 m has mass 3 kg. Density of water is 1 g/cm³. Will it float?

Volume = 0.2×0.2×0.1 = 0.004 m³ = 4000 cm³. Mass = 3 kg = 3000 g. Density = 3000/4000 = 0.75 g/cm³ < 1, so it floats.

What is the formula triangle for speed, distance, and time?

Distance on top, Speed and Time on bottom. Distance = Speed × Time; Speed = Distance / Time; Time = Distance / Speed.

What are typical units for speed?

Metres per second (m/s) or kilometres per hour (km/h).

What are typical units for density?

Grams per centimetre cubed (g/cm³) or kilograms per metre cubed (kg/m³).

What are typical units for pressure?

Newtons per square metre (N/m²), also called Pascals (Pa).

How do you convert km/h to m/s?

Multiply by 1000 and divide by 3600 (or divide by 3.6).

How do you convert m/s to km/h?

Multiply by 3600 and divide by 1000 (or multiply by 3.6).

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